#### FDA in the Frequency Domain

Viewing Eq.(7.2) in the frequency domain, the ideal
differentiator transfer-function is , which can be viewed as
the Laplace transform of the operator (left-hand side of
Eq.(7.2)). Moving to the right-hand side, the *z* transform of the
first-order difference operator is
. Thus, in the
frequency domain, the finite-difference approximation may be performed
by making the substitution

in any continuous-time transfer function (Laplace transform of an integro-differential operator) to obtain a discrete-time transfer function (

*z*transform of a finite-difference operator).

The inverse of substitution Eq.(7.3) is

As discussed in §8.3.1, the FDA is a special case of the matched transformation applied to the point .

Note that the FDA does not alias, since the conformal mapping is one to one. However, it does warp the poles and zeros in a way which may not be desirable, as discussed further on p. below.

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